The　weak　r-coloring　numbers　wcolr(G)　of　a　graph　G　were　introduced　by　the　first　two　authors　as　a　generalization　of　the　usual　coloring　number　col(G),　and　have　since　found　interesting　theoretical　and　algorithmic　applications.　This　has　motivated　researchers　to　establish　strong　bounds　on　these　parameters　for　various　classes　of　graphs.　Let　Gp　denote　the　pth　power　of　G.　We　show　that,　all　integers　p>0　and　Δ≥3　and　graphs　G　with　Δ(G)≤Δ　satisfy　col(Gp)∈O(p⋅wcol⌈p∕2⌉(G)(Δ−1)⌊p∕2⌋);　for　fixed　tree　width　or　fixed　genus　the　ratio　between　this　upper　bound　and　worst　case　lower　bounds　is　polynomial　in　p.　For　the　square　of　graphs　G,　we　also　show　that,　if　the　maximum　average　degree　2k−2<mad(G)≤2k,　then　col(G2)≤(2k−1)Δ(G)+2k+1.　©　2019　Elsevier　B.V.